Camry starter

Could be. Next time it happens give the starter a whack with hammer, while turning the key. Nippon starters used to be notorious for contactor wear. It's a cheap repair if you do it yourself.

It is true that older NipponDenso starters are notorious for their contacts wearing out when old. That typically gave just one "tick" noise rather than the clicking noise though. From what I've seen the clicking noise is cause by dirty or loose terminals. You said that the battery and cables are fine. But, what about the terminals? Make sure to take them off and clean all of the terminals and the battery posts. If you have one of those thick, non-factory, lead terminals, they are junk. Make sure to replace them with the factory ones.

Actually it's not a good idea to hit a starter anymore. Many newer starters have ceramic magnets which can break under impact and are expensive to replace. For this reason many rebuilders won't accept a starter for core exchange if there are hammer marks on case.

it yourself.

True. There are several posts on this in the Toyota NG's and on several Toyota related web sites. Should be able to google and find it easy enough. HTH, davidj92

The problem could also be (invisible) corrosion on the battery terminals. This causes high resistance. The high current for starting is unavailable and reduces the voltage enough that the solenoid drops out and chatters. So, you can check the voltage at the solenoid while cranking. The terminals must be cleaned even if they don't look corroded. Typically this happens in the spring or fall when the temperature has changed. Also, a worn ignition switch can cause the same problem.